The DBAR problem, or the

\bar

-problem, is the problem of solving the differential equation

\bar f (z, \bar) = g(z)

for the function

f(z,\bar)

, where

g(z)

is assumed to be known and

z = x + iy

is a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

in a

domain A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...

R\subseteq \Complex

. The operator

\bar

is called the DBAR operator:

\bar = \frac \left(\frac + i \frac \right)

The DBAR operator is nothing other than the complex conjugate of the operator

\partial=\frac = \frac \left(\frac - i \frac \right)

denoting the usual differentiation in the complex

z

-plane. The DBAR problem is of key importance in the theory of

integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...

, Schrödinger operators and generalizes the

Riemann–Hilbert problem In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems ...

Citations

References

*

*{{cite arXiv, last=Konopelchenko , first=B. G., title=On dbar-problem and integrable equations, year=2000 , eprint=nlin/0002049 Integrable systems, DBAR problem